Pictures WITH words

Sum of odd numbers is a square number

One extreme is the fabulous book Proof Without Words (Geogebra version here), that prides itself on beautiful but mysterious pictures that reveal mathematical structures without any words. In the Concrete Project however, we have been focussing on being able to articulate your understanding through words.

Stage 1: sketch a predicting graph and talk through as a group why you have chosen that particular shape


A pollution scientist throws herself into sketching graphs with the students

Stage 2: Formalise the sketch using data

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Stage 3: Attempt (and struggle) to describe and explain the shape of the graph in your English lesson.


Stage 4: Sketch other graphs with similar shapes, to show non-verbal understanding of a graph with a maximum

Which one is the odd one out?
Excellent work on the scale of the graph. Talking points about unusual shapes

Stage 5: With scaffolding, try again to describe and explain the graph

Mountain graphs
Step-by-step descriptions, no real explanation.
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I have spoken to this student and (I am convinced) he completely understands the concept. However, when he writes it down the understanding is invisible.

Stage 6: After repetition, repetition, repetition, students begin to be able to describe and explain the graphs.

Warning: have the students now just memorised the correct sentences?

Stage 7: Tell 1 million people about your understanding on BBC London.

One key difference between the Maths teacher and the English teacher:

  • The Maths teacher is satisfied a pupil understands the concept if they are able to wave their hands around vaguely at the right time and draw graphs to show how variables are related. Implicit understanding hinted at.
  • The English teacher is satisfied when they are able to write or say full sentences that detail how the variables are related. Explicit understanding stated.

Is this a difference between individual teachers’ preferences, or something deeper about types of understanding in different subjects?

Why is describing a graph so difficult?

A graph is a visual way of linking two variables together – how the movement of one affects the movement of another. The combining of two separate concepts (distance and air pollution for example) requires sufficient working memory and multi-step thinking. A single point on a graph represents two points on separate scales. Students who would be able to deal with one scale might struggle to make the leap to two scales. Students who understand intuitively how changing one variable affects another may therefore struggle to sketch the correct graph.

Graphs have been used to represent data in a way that we would recognise only since the early 19th century. The idea of using one point on a graph to represent multiple numbers only started with Descartes in the 17th century.  The late blooming of graphs as a way of thinking in the history of maths is a sensible pointer to the fact that conceptually they are a tricky beast.




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